To determine what both functions [tex]\( f(x) = 5e^{x+5} - 5 \)[/tex] and [tex]\( g(x) = 0.5(x-5)^2 - 5 \)[/tex] have in common, we can analyze each aspect mentioned in the options:
1. End Behavior:
- [tex]\( f(x) = 5e^{x+5} - 5 \)[/tex]: As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -5 \)[/tex] because [tex]\( e^{x+5} \to 0 \)[/tex]. As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
- [tex]\( g(x) = 0.5(x-5)^2 - 5 \)[/tex]: As [tex]\( x \to \pm\infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex] because the [tex]\( (x-5)^2 \)[/tex] term dominates.
They do not have the same end behavior.
2. Vertical Stretch:
- Vertical stretch refers to the coefficient that stretches the graph vertically relative to the base function.
- For [tex]\( f(x) = 5e^{x+5} - 5 \)[/tex], the stretch factor is 5.
- For [tex]\( g(x) = 0.5(x-5)^2 - 5 \)[/tex], the stretch factor is 0.5.
They do not have the same vertical stretch.
3. Horizontal Translation:
- Horizontal translation refers to how much the graph is shifted left or right.
- For [tex]\( f(x) = 5e^{x+5} - 5 \)[/tex], the exponent [tex]\( x+5 \)[/tex] shows a horizontal shift to the left by 5 units.
- For [tex]\( g(x) = 0.5(x-5)^2 - 5 \)[/tex], the term [tex]\( (x-5)^2 \)[/tex] shows a horizontal shift to the right by 5 units.
They do not have the same horizontal translation.
4. Vertical Shift:
- Vertical shift refers to how much the graph is shifted up or down.
- Both [tex]\( f(x) = 5e^{x+5} - 5 \)[/tex] and [tex]\( g(x) = 0.5(x-5)^2 - 5 \)[/tex] have a [tex]\(-5\)[/tex] constant term, indicating that both functions are shifted downward by 5 units.
They have the same vertical shift.
Therefore, the correct answer is:
[tex]\[ D. \text{They have the same vertical shift.} \][/tex]
